The local theta correspondence and the local Gan–Gross–Prasad conjecture for the symplectic-metaplectic case

We prove the local Gan–Gross–Prasad conjecture for the symplectic-metaplectic case under some assumptions. This is the last case of the local Gan–Gross–Prasad conjectures which has not been established. We also prove two of Prasad’s conjectures on the local theta correspondence in the almost equal rank case.